On the effect of shape parameterization on aerofoil shape optimization

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 04 Issue: 02 | Feb-2015, Available @ http://www.ijret.org 123
ON THE EFFECT OF SHAPE PARAMETERIZATION ON AEROFOIL
SHAPE OPTIMIZATION
Kailash Manohara Selvan1
1
Formerly at Hindustan University, Department of Aeronautical Engineering, Chennai, India
Abstract
The selection of suitable shape parameterization technique is one of the significant factors affecting the fidelity of the solution
found during aerofoil shape optimization process. This paper investigates the effect of shape parameterization on an automated
aerofoil shape optimization problem. Four well known shape parameterization techniques were considered for study; Bezier
curves, Class-Shape function Transformation, Hicks-Henne “Bump” function and polynomial method. A boundary layer panel
code was coupled with a surrogate-based multi-objective evolutionary algorithm and implemented within an automatic design
loop. The optimization problem was formulated for NACA 0012 aerofoil at 5 degree angle of attack. The main criteria for
comparison were based on the number of parameters required by each method for accurate representation of the aerofoil, the
ability to find the aerofoil with the best performance within the constrained design space and also the computational cost.
Preliminary results show that the optimization process was able to increase the lift-to-drag ratio of the aerofoil by 30%. Class-
Shape Transformation and Hicks-Henne Bump function were able to find the best aerofoil shape within the design space
effectively.
Keywords—Aerofoil Optimization, Evolutionary Algorithm, Shape Parameterization
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1. INTRODUCTION
n the recent years, aerodynamic shape optimization has
attracted extensive research interest. Mathematically
representing the aerofoil shape during the optimization
process is a difficult problem. Shape parameterization deals
with representing the aerofoil shape in terms of influential
design parameters. There exist many techniques for
parameterization, each requiring different number of design
parameters to represent an aerofoil. More the number of
parameters, the optimization process could be capable to
find novel aerofoil shapes. On the other hand, with more
design parameters, it would be expensive to search the
design space. In addition, the interaction between the
parameters may lead to un-realistic aerofoil being produced
[1].
Aerofoils can be represented as coordinates points. This
method can be capable to represent a variety of aerofoils and
also reflect precise local changes. However, it would be very
difficult to use coordinate based method within an
optimization process, because it would involve enormous
number of parameters and the associated high computation
cost in exploring the design space. Another method of
representing the aerofoil is in the form of B-Splines. Some
of the disadvantages of B-Spline method are presented in
[1]. PARSEC technique is another popular parameterization
technique. The comparison of coordinate method, B-Spline
curves and PARSEC method for optimization and inverse
design problem is presented in [1],[2] and [3].
This paper studies the effect of four shape parameterization
methods on optimization of a subsonic aerofoil. The
methods studied are Bezier curves, Class-Shape function
Transformation (CST), Hicks-Henne “Bump” (HHB)
function and Polynomial method. The paper is organized as
follows; the different parameterization techniques are
described in the next section, which is followed by the
description of the framework that was developed for the
research. The problem setup and the results are presented
and discussed in the subsequent sections.
2. AEROFOIL SHAPE PARAMETERIZATION
2.1 Class-Shape function Transformation (CST)
Technique
This technique proposed by Kulfan and Bussoletti [4]
represents the shape of a two-dimensional aerofoil in terms
of a shape function , class function and trailing
edge thickness .
(5)
The class function is given by,
(6)
Where and define a specific aerofoil class. Different
aerofoil class exists including elliptical ( ),
Sears-Haack body ( , biconvex (
) etc. For NACA type aerofoils with rounded leading edge
and blunt trailing edge and .
In principle, the shape function can be arbitrary. However, it
is convenient to choose a family of well behaved analytical
functions, to generate , so that the class function ,
I

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should remain as the only source of non-analyticity of the
representation [4]. A weighted sum of Bernstein
polynomials can be used as shape function, Eq (7)
(7)
For and , the shape function at the
extremes can be related to leading edge radius ( ) and
trailing edge thickness ( ) and angle ( ) as follows,
(8)
Fig 1 shows the parametric aerofoil generated using CST
technique; (a) shows the Class function with and
, (b) shows the Shape function with 1st
order and 3rd
order Bernstein polynomials and (c) shows the parametric
aerofoil.
2.2 Hicks-Henne “Bump” Function
This technique proposed by Hicks and Henne [5][6], uses a
set of smooth function to perturb the initial aerofoil. The
aerofoil is represented as a weighted sum of sine function.
(1)
(2)
and define the maximum thickness location of the
bump and the width of the bump respectively. denote the
shape parameters. An important advantage of this method is
that it allows specific region of the aerofoil to be perturbed
while maintaining the others virtually undisturbed.
Fig 2 shows an aerofoil represented through HHB technique
with 2 sine functions ( )
2.3 Bezier
A Bezier curve is a parametrically defined curve given by,
(3)
is the vector of control points of the parametric curve
. The blending function is the Bernstein polynomial
given by,
(4)
The Bezier curve can be used to parameterize any arbitrary
curve and therefore they can be used for a variety of
applications including aerofoil optimization.
Fig 6 shows an aerofoil represented using a Bezier curve
with 6 control points.
2.4 Polynomial Function
The aerofoil shape can be represented by higher order
polynomials, Eq (9).
(9)
The number of parameters is defined by the order of the
polynomial chosen. PARSEC method [7] is a kind of
polynomial model which uses 11 parameters to represent the
aerofoil.
(10)
However, in this paper a generic polynomial function is
considered.
A simulation framework Project X-2D was developed as
part of this research. A brief description about the
framework is presented in the next section.
3. PROJECT X - 2D
Project-X 2D comprises a set of program codes specifically
developed for automatic design and optimization of two-
dimensional aerofoils. The illustration of Project-X 2D
framework is presented in Fig 5.
The user specifies the reference aerofoil. The reference
aerofoil is parameterized by the Para_X module by one of
the techniques described in the previous section. The user
then defines the design space by specifying the
bounds/constraints for each shape parameter.
Conventionally, aerofoils will be generated by randomly
picking parameters from the defined design space. In the
present work, aerofoils are not generated randomly; rather,
they are generated using parameters selected by Design of
Experiments (DoE) technique. Latin-Hypercube Sampling
(LHS) technique is implemented within this framework.
LHS uses a stratified sampling scheme to improve on the
coverage of the input design space. An array of aerofoil
shapes are generated and stored by the DoE_X module.
The generated aerofoils are solved by Xfoil [8]. XFOIL is a
design and analysis system for isolated subcritical airfoils. A
linear-vorticity second order accurate panel method is used
in the inviscid flow. This method is coupled with an integral
boundary-layer method and an -type transition
amplification formulation using a global Newton method.
XFOIL was chosen for use in Project X-2D based on its
speed and accuracy. It is very fast compared with Reynolds-
averaged Navier-Stokes (RANS) methods, and it has been
proven to be well suited for the analysis of subsonic airfoils
even at low Reynolds numbers [9].

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Fig 1 Shape parameterization using CST technique with 1st
order and 3rd
order shape function
Fig 2 Shape parameterization using HHB technique ( and )

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Fig 3 Shape parameterization using Bezier technique with 6 control points
Fig 4 Shape parameterization using Polynomial technique (2nd
order and 6th
order)

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Fig 5 Project-X 2D Framework
Xfoil provides the aerodynamic performance (lift and drag
coefficients) for each aerofoil generated by DoE_X module.
The design parameters for the initial population and their
corresponding aerodynamic data are stored as a database by
Database_X module.
A surrogate (approximate) model can be constructed which
closely mimics the relationship between the input geometric
parameters and output performance variables. This
approximate model can be used during the optimization
process. The total number of flow solver calls can be
noticeably reduced through this technique, which in turn
reduces the total time without deteriorating the performance
of the optimization process [10]. Response surface
approximations (RSA) are often used as inexpensive
replacements for computationally expensive computer
simulations. Once a RSA has been computed, it is cheap to
evaluate the objective functions during optimization. So a
RSA based surrogate model is implemented within this
framework.
Evolutionary Algorithm (EA) or Genetic Algorithm (GA) is
very popular for engineering design optimization. They
were originally developed using natural evolution analogy.
They facilitate the set of designs “offsprings” with improved
performance or “fitness” to be evolved, through GA
operators; crossover, mutation and selection of the best
designs from the previous generation of. designs “parents”.
This algorithm is renowned for its ability to seek the global
optimum and the ability to handle multiple objectives
effectively. A Multi-Objective Genetic Algorithm,
MOGA_X module was developed and implemented within
this framework.
The MOGA_X module uses the approximate model during
design space exploration. The optimizer performs a
predefined number of generations and the flow solver is
called only for the optimal design found by this inner
optimization loop. This design is added into the database
with an intention to improve the accuracy of the
approximate model for the next design/optimization cycle.
The robustness of this methodology is affected by the
stability of the surrogate model and the construction of a

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stable approximate model is essential. This methodology is
expected to converge much faster than conventional
optimization methods. A predefined number of design
iterations are performed before producing the optimal
aerofoil shape.
The problem setup and the results are presented in the next
section.
4. RESULTS
4.1 Problem Setup
Any optimization problem needs initial reference geometry
to start with. In the present study, NACA 0012 aerofoil was
specified as the reference. The next step is specifying the
bounds through which the aerofoil shape could be perturbed
during the optimization process. Mechanical, structural
considerations might limit the design space. In this case the
design space was constrained within the minimum and
maximum bounds specified. The bounds and the reference
aerofoil are shown in Fig 6.
Fig 6 Reference Aerofoil (NACA 0012) with minimum and maximum bounds specified
Different parameterization techniques were used to
parameterize the reference aerofoil. The suction and
pressure surfaces were parameterized separately. LHS
method was used to select the initial population from the
design space. The effect of the initial sample size using LHS
on the accuracy of the response surface constructed was
investigated in [11]. They confirmed that the initial sample
size had very little effect on the accuracy of the response
surface model constructed using LHS method. Initial
population size was selected as 2-3 times the number of the
parameters in each case. The initial population of aerofoils
was solved using Xfoil to obtain the aerodynamic
performance characteristics.
The flow conditions used by the solver are presented below
in Table 1.
Table 1 Inlet flow conditions
Flow conditions
Mach Number 0.2
Reynolds Number
Angle of attack
A response surface approximation was constructed on this
data. This model was used by the optimizer for objective
function evaluations within the inner optimization loop. The
flow solution is computed only on the final shape found at
the end of a design loop. This new flow solution is added
into the database and used for constructing the response
surface model for the next design loop. 20 generations were
allowed within each inner optimization loop and 20 design
cycles were carried out. All simulations were carried out on
an Intel dual core machine with 2GB RAM.
4.2 Lift Maximization
An optimization problem can have many objectives.
Usually, the multiple objectives can be reduced to a single
objective. This is done by assigning them some weighting
factors, and then use the single objective during
optimization.
The goal of this optimization problem was to find the
aerofoil shape from the constrained design space with the
same drag coefficient as the reference aerofoil but with
higher lift coefficient. The lift and drag coefficients of the

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reference aerofoil for the flow conditions shown in Table 1
were, and . The
objective function formulated for this optimization problem
is given in Eq (11).
(11)
4.3 Accuracy
The reference aerofoil was parameterized using all the four
techniques. The number of parameters required by each of
these techniques to adequately represent the suction surface
of the aerofoil was studied. The error in parameterization
was computed by Eq (12). Parameterizations with error less
than were accepted.
(12)
Fig 7 presents the change in error with the number of
parameters for each parameterization technique. It shows
that, HHB and CST technique require the least number of
parameters; they require 4 parameters to adequately
represent the suction surface of the reference aerofoil, while
polynomial and Bezier techniques require 7 and 12
parameters respectively. Twice as much parameters were
required to represent the complete aerofoil.
Fig 7 Comparison of number of parameters and error associated with different parameterization techniques
Table 2 Number of parameters required by each
parameterization technique to represent the complete
aerofoil
Cases Number of parameters
Class-Shape function 8
Hicks-Henne function 8
Polynomial 14
Bezier 24
4.4 Computational Cost
More parameters during parameterization imply a bigger
initial population. It is obvious that the computation time for
initial database generation increases linearly with the
number of parameters. In addition to this, the number of
parameters also influence the time for approximate model
construction and optimization. The time required per design
cycle when using different parameterization techniques are
presented in Table 3.
Table 3 Comparison of time per design loop for different
parameterization techniques
Cases Time per design
loop[s]
Class-Shape function 97
Hicks-Henne function 92
Polynomial 148
Bezier 210

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4.5 Optimum Designs
The lift coefficients and the lift-to-drag ratio of the
optimum aerofoils found by the optimizer through different
parameterization techniques are presented in Table 4. It can
be seen that the HHB and CST parameterization technique
was able to improve the lift-to-drag ratio considerably
compared to Bezier and polynomial methods. The optimum
aerofoil shapes and the corresponding coefficient of pressure
(Cp) distribution are presented in Fig 8 to Fig 11. The
optimum aerofoils found by HHB and CST techniques have
more negative pressure on the suction side and more
positive pressure on the pressure side which are key for
higher lift. Clearly, for the same design space and with the
same optimization strategy, HHB and CST facilitate
aerofoils with better aerodynamic performance to be found.
Table 4 Comparison of the performance of optimum
aerofoils found through different parameterization
techniques
Cases
Class-Shape function 0.7087 82.407 26.55357
Hicks-Henne function 0.7272 84.558 29.85714
Bezier 0.6352 73.860 13.42857
Polynomial 0.6263 72.441 11.83929
5. CONCLUSION
Four shape parameterization techniques; Class-Shape
function Transformation, Hicks-Henne “Bumps” function,
Bezier and polynomial method, are compared for the
aerodynamic design and optimization of aerofoils. A
framework is developed to facilitate this study. The lift
maximization optimization problem was formulated on
NACA 0012 reference aerofoil at 5 degree angle of attack.
Many important observations were made. Firstly, HHB and
CST technique required only 8 parameters to accurately
represent the aerofoil. Secondly, HHB and CST techniques
with lesser parameters needed atleast 50% lesser
computational time compared to the other techniques.
Thirdly, the optimum aerofoil found through CST and HHB
aerofoil showed better aerodynamic performance,
confirming their better exploratory characteristics. The
polynomial method, even if it was able to represent the
aerofoil with fewer parameters compared to the Bezier was
not able to provide the best aerofoil. It can be concluded that
CST to HHB techniques could be very effective in aerofoil
shape optimization compared to conventional techniques.
The frame work developed could be easily extended to 3D
wing design and optimization. The future work would
involve the investigation of these parameterization
techniques on an inverse design problem.
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[7] H.Sobieczky, "Parametric airfoils and wings," Notes
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[9] B. A. Gardner and Michael.S.Selig, "Aerofoil design
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[10] L.P.Swiler, R.Slepoy and A.A.Giunta, "Evaluation of
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Approximations," Sandia National Laboratories,
Albuquerque, NM.

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Fig 8 Optimum aerofoil geometry found through HHB parameterization technique and corresponding Cp distribution compared
with the reference
Fig 9 Optimum aerofoil geometry found through CST parameterization technique and corresponding Cp distribution compared
with the reference

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Fig 10 Optimum aerofoil geometry found through Bezier parameterization technique and corresponding Cp distribution compared
with the reference
Fig 11 Optimum aerofoil geometry found through Polynomial parameterization technique and corresponding Cp distribution
compared with the reference

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